The Serbian winger has signed a five-year contract with the club after the Swabians were relegated from the Bundesliga last campaign.
Despite interest from Liverpool, Kostic caught the eye of Hamburg after the player became available – joining his fellow teammate Arianit Ferati who also joined the club from Stuttgart.
The club's director, Dietmar Beiersdorfer, was delighted with the signing, stating; "He's a player whose actions are at a high tempo, who always looks for one-on-one situations. Despite being only 23, he has a lot of experience, and has proven his qualities in the last two years."
The 23-year-old played 59 games for Stuttgart in the Bundesliga during his two year spell, scoring eight and assisting 13 goals.
HSV not the only team interested in the Serbian
Kostic joins Alen Halilovic at the club after the Croatian joined in a €5 million move from FC Barcelona on a four-year contract.
HSV reportedly beat Liverpool to the signature of Kostic, with Jürgen Klopp said to be interested in signing the 23-year-old, who instead opted to stay in Germany, with the Reds instead sealing a more expensive move for Sadio Mane.
How have Hamburg fared in pre-season?
Bruno Labbadia and Bernhard Peters have both been praised for their work this summer, which has seen Bobby Wood and Luca Waldschmidt join the club from Union Berlin and Eintracht Frankfurt respectively.
It’s likely that the HSV coach will start with his favoured 4-2-3-1 formation, which would likely see Kostic on the left – where he has played predominately for Stuttgart – and Halilovic on the right, whilst Aaron Hunt or Michael Gregoritsch in behind the striker.